I was awarded a PhD in mathematics by the University of Copenhagen in July 2022 on the basis of my dissertation titled On the Hochschild Homology of Hypersurfaces as a Mixed Complex. You can download different versions further below.

Abstract

In this thesis we describe Hochschild homology over π‘˜ of quotients of polynomial algebras π‘˜[π‘₯₁,…,π‘₯β‚™] / 𝑓 for certain polynomials 𝑓 in 𝑛 ≀ 2 variables, as an object of the ∞-category of mixed complexes ℳ𝑖π‘₯𝑒𝑑, where π‘˜ is a commutative ring in which 2 is invertible.

In 1992, the Buenos Aires Cyclic Homology Group [BACH] constructed, for any n and any commutative ring π‘˜, a quasiisomorphism between the standard Hochschild complex over π‘˜ of π‘˜[π‘₯₁,…,π‘₯β‚™] / 𝑓 and a quite small chain complex, under the assumption that 𝑓 is monic with respect to a chosen monomial order. This result was improved upon by Larsen in 1995 [Larsen] by taking the mixed structure into account as well, though only considering polynomials 𝑓 in 𝑛 = 2 variables that are monic with respect to one of the variables.

Assuming a conjectural description of Hochschild homology of polynomial rings, we extend these previous results by constructing, for a large subset of the polynomials 𝑓 considered in [BACH], a strict mixed structure on the chain complex described in [BACH] and showing that it represents the Hochschild homology over π‘˜ of π‘˜[π‘₯₁,…,π‘₯β‚™] / 𝑓 as an object in the ∞-category of mixed complexes. We also verify the conjecture in some cases, leading to unconditional results for 𝑛 ≀ 2 variables, as long as 2 is invertible in π‘˜.

The results of this thesis do not rely on the two aforementioned prior results, but instead use the modern approach to Hochschild homology based on ∞-categorical methods. Along the way, to be able to state and prove our result in this setting, we prove some results that may be of independent interest.

[BACH] Jorge Alberto Guccione, Juan Jose Guccione, Maria Julia Redondo, and Orlando Eugenio Villamayor. β€œHochschild and Cyclic Homology of Hypersurfaces”. In: Advances in Mathematics 95.1 (1992), pp. 18–60. ISSN: 0001-8708. DOI: 10.1016/0001-8708(92)90043-K.

[Larsen] Michael Larsen. β€œFiltrations, Mixed Complexes, and Cyclic Homology in Mixed Characteristic”. In: K-Theory 9 (1995), pp. 173–198. DOI: 10.1007/BF00961458.

Full text

Link Date Description
pdf (679 pages) 2022-04-12 Version I submitted, for reading on a screen, A4 format.
Volume 1 pdf (366 pages)
Volume 2 pdf (390 pages)		 2022-06-17 Version that was printed, in two volumes, B5 format. Includes changes in layout and typography as well as corrections for some typos.